Questions: Three-circle, red-on-white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at an archaeological site. Suppose that at one excavation, a sample of 161 potsherds indicated that 70 were of the three-circle, red-on-white pattern. (a) Find a point estimate p^- for the proportion of all ceramic potsherds at this site that are of the three-circle, red-on-white pattern. (Round your answer to four decimal places.) (b) Compute a 95% confidence interval for the population proportion ρ of all ceramic potsherds with this distinctive pattern found at the site. (Enter your answer in the form: lower limit to upper limit. Include the word "to." Round your numerical values to three decimal places.)

Three-circle, red-on-white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at an archaeological site. Suppose that at one excavation, a sample of 161 potsherds indicated that 70 were of the three-circle, red-on-white pattern. (a) Find a point estimate p^- for the proportion of all ceramic potsherds at this site that are of the three-circle, red-on-white pattern. (Round your answer to four decimal places.) (b) Compute a 95% confidence interval for the population proportion ρ of all ceramic potsherds with this distinctive pattern found at the site. (Enter your answer in the form: lower limit to upper limit. Include the word "to." Round your numerical values to three decimal places.)
Transcript text: Three-circle, red-on-white is one distinctive pattern painted on ceramic vessels of the Anasazi period found at an archacological site. Suppose that at one excavation, a sample of 161 potsherds indicated that 70 were of the three-circle, red-on-white pattern. (a) Find a point estimate $p^{-}$for the proportion of all ceramic potsherds at this site that are of the three-circle, red-on-white pattern. (Round your answer to four decimal places.) (b) Compute a $95 \%$ confidence interval for the population proportion $\rho$ of all ceramic potsherds with this distinctive pattern found at the site. (Enter your answer in the form: lower limit to upper limit. Include the word "to." Round your numerical values to three decimal places.)
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Solution

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To solve the given problem, we need to address two parts: finding a point estimate for the proportion and computing a 95% confidence interval for the population proportion.

(a) Point Estimate for the Proportion:

The point estimate for the proportion of potsherds with the three-circle, red-on-white pattern is calculated using the formula for the sample proportion:

\[ \hat{p} = \frac{x}{n} \]

where \( x \) is the number of potsherds with the pattern, and \( n \) is the total number of potsherds in the sample.

Given:

  • \( x = 70 \)
  • \( n = 161 \)

\[ \hat{p} = \frac{70}{161} \approx 0.4348 \]

So, the point estimate \( \hat{p} \) is 0.4348.

(b) 95% Confidence Interval for the Population Proportion:

To compute the 95% confidence interval for the population proportion, we use the formula:

\[ \hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]

where \( Z \) is the Z-score corresponding to the desired confidence level. For a 95% confidence interval, \( Z \approx 1.96 \).

First, calculate the standard error (SE):

\[ SE = \sqrt{\frac{0.4348 \times (1 - 0.4348)}{161}} \approx \sqrt{\frac{0.4348 \times 0.5652}{161}} \approx \sqrt{\frac{0.2458}{161}} \approx 0.0388 \]

Now, calculate the margin of error (ME):

\[ ME = 1.96 \times 0.0388 \approx 0.0760 \]

Finally, compute the confidence interval:

\[ \text{Lower limit} = 0.4348 - 0.0760 = 0.3588 \] \[ \text{Upper limit} = 0.4348 + 0.0760 = 0.5108 \]

Thus, the 95% confidence interval for the population proportion is:

0.359 to 0.511 (rounded to three decimal places).

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